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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002
The Design of Approximate Hilbert Transform
Pairs of Wavelet Bases
Ivan W. Selesnick, Member, IEEE
Abstract—Several authors have demonstrated that significant
improvements can be obtained in wavelet-based signal processing
by utilizing a pair of wavelet transforms where the wavelets form
a Hilbert transform pair. This paper describes design procedures,
based on spectral factorization, for the design of pairs of dyadic
wavelet bases where the two wavelets form an approximate Hilbert
transform pair. Both orthogonal and biorthogonal FIR solutions
are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design
procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given,
and examples are also given to illustrate the results.
the scaling filters should be offset by a half sample. In [18], a
design problem was formulated for the minimal length scaling
filters such that 1) the wavelets each have a specified number of
, and 2) the half-sample delay approxvanishing moments
with specified degree
. However,
imation is flat at
this formulation leads to nonlinear design equations, and the examples in [18] had to be obtained using Gröbner bases. In this
paper, we describe a design procedure based on spectral factorization. It results in filters similar to those of [18]; however, the
design algorithm is much simpler and more flexible.
Index Terms—Dual-tree complex wavelet transform, Hilbert
transform, wavelet transforms.
A. Preliminaries
represent a CQF pair [22]. That is
Let the filters
I. INTRODUCTION
T
HIS paper describes design procedures, based on spectral
factorization, for the design of pairs of dyadic wavelet
bases where the two wavelets form an approximate Hilbert
transform pair. Several authors have advocated the simultaneous use of two wavelet transforms where the wavelets
are so related. For example, Abry and Flandrin suggested
using a Hilbert pair of wavelets for transient detection [2] and
turbulence analysis [1]. Ozturk et al. suggested it for waveform
encoding [14]. They are also useful for implementing complex
and directional wavelet transforms. Freeman and Adelson
employ the Hilbert transform in the development of steerable
filters [5], [20]. Kingsbury’s complex dual-tree DWT [8], [9] is
based on (approximate) Hilbert pairs of wavelets. The steerable
pyramid and the dual-tree DWT have numerous benefits,
including improved denoising capability and the fact that they
are both directional and nearly shift-invariant. The paper by
Beylkin and Torrésani [3] is also of related interest.
One could start with a known wavelet and then take its Hilbert
transform to obtain the second wavelet; however, in that case,
the second wavelet would not be of finite support. One could
design a finitely supported wavelet to approximate the infinitely
supported Hilbert transform, but in this paper, we design both
wavelets together to better utilize the degrees of freedom.
Using the infinite product formula, it was shown in [18] that
for two orthogonal wavelets to form a Hilbert transform pair,
Manuscript received December 6, 2000; revised January 7, 2002. This work
was supported in part by the National Science Foundation under CAREER Grant
CCR-987452. The associate editor coordinating the review of this paper and
approving it for publication was Dr. Ta-Hsin Li.
The author is with the Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: [email protected]).
Publisher Item Identifier S 1053-587X(02)03148-3.
and
Equivalently, in terms of the
, where
is an odd integer.
-transform, we have
and
Let the filters
represent a second CQF pair. In this
are real-valued filters. It is convepaper, we assume
nient to write the CQF condition in terms of the autocorrelation
functions defined as
or equivalently as
Then,
if
and
satisfy the CQF conditions if and only
are halfband filters:
and similarly for
. This can be written more compactly as
and
1053-587X/02$17.00 © 2002 IEEE
and
(1)
SELESNICK: DESIGN OF APPROXIMATE HILBERT TRANSFORM PAIRS OF WAVELET BASES
The dilation and wavelet equations give the scaling and wavelet
functions
(2)
(3)
and wavelet
are defined simiThe scaling function
and
.
larly but with filters
is denoted by
.
Notation: The -transform of
will be denoted
The discrete-time Fourier transform of
, although it is an abuse of notation. The Fourier
by
is denoted by
.
transform of
B. Hilbert Transform Pairs
and
In [18], it was shown that if
CQF filters with
for
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or equivalently
around
The coefficients
in (5) can be computed very efficiently
using the following ratio.
From this ratio, it follows that the filter
as follows:
can be generated
are lowpass
(4)
This can be implemented in Matlab with the commands
then the corresponding wavelets form a Hilbert transform pair
That is
In our problem, we will use
, we have
example, with
in (5) with
. For
for
C. Flat-Delay Allpass Filter
The design procedure presented in this paper depends on the
design of an allpass filter with approximately constant fractional
delay. Several authors have addressed the design of allpass systems that approximate a fractional delay [11], [12], [16]. The
following formula for the maximally flat delay allpass filter
is adapted from Thiran’s formula for the maximally flat delay
allpole filter [23]. The maximally flat approximation to a delay
of samples is given by
With
, we have
for
II. ORTHOGONAL SOLUTIONS
In this section, we look for pairs of orthonormal wavelets
where the lowpass scaling filters have the form
where the filter
will be chosen to achieve the (approximate)
half-sample delay. The first step of the design procedure will be
to achieve the desired
to determine the appropriate filter
and
. In terms of the transfer
relationship between
functions, we have
where
with
(5)
where
represents the rising factorial
and
have the common divisor
determined later. We can write
, which will be
where we can recognize that the transfer function
With this
, we have the approximation
around
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is an allpass system
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002
. Therefore
TABLE I
MATLAB PROGRAM
and
If the allpass system
is an approximate half-sample delay
around
or equivalently,
around
after approximation (4) is achieved
, then the sought-
around
This is achieved by taking the allpass filter
in (5) with
. (The point
is chosen for the point of approximation
for the lowpass filter because that is the center of the passband.)
To obtain wavelet bases with vanishing moments, we let
Then
(6)
(7)
We now have the following design problem. Given
of minimal degree such that
and
find
the CQF conditions (1). Note that with (6) and (7),
have the same autocorrelation function
and ,
satisfy
and
Similar to the way Daubechies wavelet filters are obtained, we
using a spectral factorization approach as in
can obtain
[22]. The procedure consists of two steps.
of minimal length such that
1) Find
, and
a)
is halfband.
b)
of minimal length will be supported on the
Note that
.
range
to be a spectral factor of
2) Set
(8)
To carry out the first step, we need only solve a system of linear
equations. Defining
we can write the halfband condition as
When written in matrix form, this calls for a square matrix of
, which has the form of a convolution
dimension
(Toeplitz) matrix with every second row deleted.
The second step assumes
permits spectral factorization,
which we have found to be true in all our examples. With
obtained in this way, the filters
and
defined in (6)
and (7) satisfy the CQF conditions and have the desired halfis not unique.
sample delay. Note that
and
of
This design procedure yields filters
. A Matlab program to imple(minimal) length
ment this design procedure is given Table I. The commands
binom and sfact for computing binomial coefficients and
performing spectral factorization are not currently standard
Matlab commands. They are available from the author.
and
, the filters
and
Example 1A: With
are of length 12. Fig. 1 illustrates the solution obtained
from a mid-phase spectral factorization. The plot of the function
shows that it approximates zero for
, as
and
form a Hilbert transform pair. The coefexpected if
ficients are given in Table II. The Sobolev exponent (reflecting
the smoothness of the wavelets) was found1 to be 1.983. Note
and
.
that the Sobolev exponent is the same for
and
again, but this
Example 1B: We set
time, we take a minimum-phase spectral factor rather than a
mid-phase one. The wavelets obtained in this case are illustrated
is exactly the same
in Fig. 2. The function
as in Example 1A; using a different spectral factor in (8) does
not change it. We will see below that the mid-phase and minimum-phase solutions can lead to different results when they are
used to implement two-dimensional (2-D) directional wavelet
transforms. (The Sobolev exponent is the same as for Example
1A.)
and
, the filters
and
Example 2: With
are again of length 12. Fig. 3 illustrates a solution using a
mid-phase spectral factor. It can be seen that
1The Sobolev exponents were computed using the Matlab program sobexp
by Ojanen (http://www.math.rutgers.edu/~ojanen/).
SELESNICK: DESIGN OF APPROXIMATE HILBERT TRANSFORM PAIRS OF WAVELET BASES
1147
Fig. 2. Example 1B. Approximate Hilbert transform pair of orthonormal
wavelet bases with N = 12; K = 4; L = 2; and minimum-phase spectral
factorization.
Fig. 1. Example 1A. Approximate Hilbert transform pair of orthonormal
wavelet bases with
12; K = 4; L = 2; and mid-phase spectral
factorization.
N =
TABLE II
COEFFICIENTS FOR EXAMPLE 1A AND EXAMPLE 2
Fig. 3. Example 2. Approximate Hilbert transform pair of orthonormal
wavelet bases with N = 12; K = 3; L = 3;, and mid-phase spectral
factorization.
A. Directional 2-D Wavelets
One of the important applications of a Hilbert pair of wavelet
bases is the implementation of directional 2-D (overcomplete)
wavelet transforms, as illustrated in [7]. The directional
wavelets are obtained by first defining a 2-D separable wavelet
basis via
is closer to zero for negative frequencies. At the same time, the
Sobolev exponent decreases to 1.736. This is to be expected,
as we have reduced the number of vanishing moments and at
the same time increased the degree of approximation for the
half-sample delay. The coefficients are given in Table II.
In addition, define
by
similarly. Then, the six wavelets defined
(9)
(10)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002
or equivalently
Combining with (4), we get the condition
or
Therefore
(11)
Fig. 4. Two-dimensional wavelets generated by an approximate Hilbert
transform pair of 1-D wavelets, using a mid-phase spectral factor (Example
1A) and a minimum-phase spectral factor (Example 1B), respectively.
has linear phase, and the delay is offset by
In this case,
one quarter of a sample from the center of symmetry of a length
filter
. As a CQF filter can not have exact linear phase,
the filters given in [9] approximately satisfy (11). The design of
orthonormal wavelets with approximate linear phase with noninteger delay has also been described in [13], [19], and [25]. An
alternative, simple, way to obtain an approximate Hilbert transform pair of (biorthogonal) wavelets with approximate linear
phase is to modify the spectral factorization approach, as described in the next section.
III. BIORTHOGONAL SOLUTIONS
for
are directional, as illustrated in Fig. 4. A wavelet
transform based on these six wavelets can be implemented by
taking the sum and difference of two separable 2-D DWTs. The
resulting directional wavelet transform is two-times redundant.
The inverse requires taking the sum and difference, dividing by 2,
and the separable inverse DWTs. The four-times redundant DWT
presented by Kingsbury is both directional and complex [8].
We note that the type of spectral factorization performed in
the design procedure described above influences the quality of
the directionality of the 2-D wavelets. For example, the wavelets
of example 1A, which were constructed with a mid-phase spectral factor, lead to the six 2-D wavelets shown in the top panel
of Fig. 4. On the other hand, the wavelets of example 1B, which
were constructed with a minimum-phase spectral factor, lead to
the six 2-D wavelets shown in the lower panel of Fig. 4. In this
case, the directionality is not as clean as in the first case. Instead,
some curvature is present. The figure shows that the mid-phase
spectral factor can be preferable to the minimum-phase one for
the implementation of directional wavelet transforms.
Evidently, better directional selectivity is obtained when the
two wavelets have approximate (or exact) linear phase in addition to forming an approximate Hilbert transform pair. The
procedure described above imposes a condition on the phase of
, but it does not impose any condition of the
of
and
individually. To ensure a high diphases of
and
rectional selectivity, one could impose directly that
have approximately linear phase rather than relying on
the near linear-phase of a mid-phase spectral factor. For example, consider the system described in [9]. In [9], the filters
and
are related through a reversal
The quality of the directional 2-D wavelets derived from a
pair of 1-D wavelets appears to depend on the wavelets having
approximately linear phase, in addition to forming an approximate Hilbert transform pair. If biorthogonal wavelets are acceptable, then the procedure given in Section II can be modified to
and
have apyield wavelet bases for which both
proximately linear phase. They can then be used to generate 2-D
wavelet bases with improved directional selectivity. To generate
wavelets with approximate linear phase, we can follow the approach of [4] for the design of symmetric biorthogonal wavelets,
in which the spectral factorization of a halfband filter is replaced
by its factorization into two linear-phase, but different, filters. In
the biorthogonal case, we denote the dual scaling functions and
and
. As we have a pair of
wavelets by
biorthogonal wavelet bases, we have eight filters including the
dual filters, corresponding to the filterbanks illustrated in Fig. 5.
and wavelet
The dual scaling function
by the equations
are given
The dual scaling function
and wavelet
are similarly defined.
The goal will be to design the filters so that both the primary (synthesis) and dual (analysis) wavelets form approximate
Hilbert transform pairs
and
SELESNICK: DESIGN OF APPROXIMATE HILBERT TRANSFORM PAIRS OF WAVELET BASES
1149
where
is odd. denotes the number of vanishing modenotes the
ments of the primary (synthesis) wavelets, and
number of vanishing moments of the dual (analysis) wavelets.
and
are then given by
The product filters
(14)
(15)
Fig. 5.
To obtain the required halfband property, we find a symmetric
so that
odd-length sequence
Filterbank structures for the biorthogonal case.
If we define the product filters as
is halfband. The symmetric sequence
can be obtained by
solving a linear system of equations as in Section II. We can
and
by factoring
then obtain
then the biorthogonality conditions can be written as
(16)
(12)
(13)
must both be halfband. With out loss of
That is to say,
generality, we can assume that is an odd integer. In that case,
the highpass filters are given by the following expressions [4],
[24]:
and
are symmetric. As
and
where both
are symmetric, so are
and
. It follows that
and
are related by a reversal
(17)
and similarly
(18)
We will look for solutions of the form
The problem is to find
and
such that we have have
the following.
1) The biorthogonality conditions (12) and (13) are satisfied.
vanishing moments ( vanishing
2) The wavelets have
moments for the dual wavelets).
3) The wavelets form an approximate Hilbert transform pair.
To obtain the half sample delay needed to ensure the approxto come from the flat
imate Hilbert property, we choose
delay allpass filter, as before:
around
To ensure the vanishing moments properties, we take
to be of the form
and
and
have approximately linear phase
Therefore,
does) in addition to satisfying (4) approximately.
(because
Note that the symmetric factorization (16) is not unique—many
solutions are available. In addition, note that the sequences
and
do not need to have the same length.
and
, we can take the
Example 3: With
and
to be of length
. The
synthesis filters
and
will then be of length 11. Fig. 6
analysis filters
illustrates this solution obtained from a symmetric factorization.
and likewise
The plots of
show that they approximate zero for
, as expected. The
and
are given in Table III. The coefcoefficients
and
are given by their reversed versions,
ficients
and
as in (17) and (18). The Sobolev exponent of
is 1.731, whereas the Sobolev exponent of
and
is
2.232.
Fig. 7 illustrates the analysis and synthesis directional 2-D
wavelets derived from the 1-D wavelets using (9) and (10).
IV. IIR SOLUTIONS
The spectral factorization approach can also be used to construct orthogonal wavelet bases based on recursive infinite imis a rational
puilse response (IIR) digital filters, where
function of . Wavelets based on rational scaling filters have
been discussed, for example, in [6], [15], [17], and [21]. IIR
filters often require lower computational complexity than finite
impulse response (FIR) filters. Analogous to the approach given
in [6], but with the flat delay filter included, approximate Hilbert
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002
TABLE III
COEFFICIENTS FOR EXAMPLE 3
Fig. 7. Two-dimensional wavelets generated by an approximate Hilbert
transform pair of biorthogonal 1-D wavelets (Example 3).
Fig. 6. Example 3: Approximate Hilbert transform pair of biorthogonal
~ = 4; L = 2.
13; K = K
wavelet bases with
N=
The product filter is given by
transform pairs of IIR wavelets can be obtained with the following form:
Defining
the orthogonality condition
as
can be written
(19)
SELESNICK: DESIGN OF APPROXIMATE HILBERT TRANSFORM PAIRS OF WAVELET BASES
1151
spectral-factorization does not always provide a solution with
a nearly linear phase response). However, the biorthogonal
solutions given in this paper have approximately linear phase.
As a result, the orthogonal solutions by Kingsbury are more
nearly linear phase than the orthogonal solutions given here;
likewise, the linear-phase biorthogonal solutions by Kingsbury
are more nearly orthogonal than the approximately linear-phase
biorthogonal solutions given here.
In addition, the design procedure developed in this paper
and the ones described by Kingsbury are quite different.
Kingsbury’s design procedures are based on minimizing the
aliasing of a filterbank after it is iterated a fixed number stages.
The minimization is performed over a parameterized space of
perfect reconstruction filterbanks using iterative optimization
algorithms. On the other hand, the approach taken in this paper
is based on the limit functions [defined by (2) and (3)] and on
vanishing moments. As a result, the limit functions
and
are closer to forming a Hilbert pair for the filters given
in this paper than those presented by Kingsbury, whereas the
solutions by Kingsbury do this better for the first several stages.
VI. CONCLUSION
Fig. 8. Example 4. Approximate Hilbert transform pair of orthonormal IIR
wavelet bases with K
2; L = 2.
=
can be found by spectral factorization; note that the
left-hand side of (19) is a function of . A stable filter is
obtained using the minimum-phase spectral factor in (19).
and
Example 4: With two vanishing moments
, we obtain the following stable causal transfer functions
Fig. 8 illustrates the two wavelets
.
tude of
and the magni-
V. COMPARISON TO KINGSBURY’S FILTERS
It is interesting to compare the filters obtained here with
those presented by Kingsbury in [7]–[10], where the complex dual-tree DWT is introduced. In [7] and [8], Kingsbury
presents linear-phase biorthogonal solutions where one scaling
function is symmetric about an integer, whereas the other
scaling function is symmetric about an integer plus 0.5. In
this case, the (approximate) half-sample delay [see (4)] can
be achieved by making the frequency responses magnitudes
. The family of
(approximately) equal
filter banks introduced in [9] and [10] are orthogonal but with
the property that one scaling function is the time-reversed
version of the other. Then, the frequency responses magnitudes
are equal, and it remains to design the phase response of the
filter to achieve (approximately) the half-sample shift [see
(11)]. On the other hand, the approach described in this paper
does not impose any condition on the phase-response of the
filters. The orthogonal solutions given in this paper do not have
approximately linear phase (for short filters the mid-phase
This paper presents simple procedures for the design of pairs
of wavelet bases where the two wavelets form an approximate
Hilbert transform pair. The approach proposed here is analogous to the Daubechies construction of compactly supported
wavelets with vanishing moments but where the approximate
Hilbert transform relation is added by way of incorporating a
flat delay filter.
The approach is based on a characterization of Hilbert transform pairs of wavelets bases given in [18]. The formulation of
the problem, using a flat delay allpass filter, makes it possible to
employ the spectral factorization design method as introduced
in [22] for the design of CQF filters. Note that even though an
allpass filter arises in the problem formulation, the filters we
obtain are FIR (IIR solutions are also available, as described in
Section IV).
Given an allpass filter, the proposed design method produces
short filters with a specified number of vanishing wavelet moments. Although a flat delay filter was used here, any other allpass filters that approximate a delay of a half sample could be
used instead. The degree of the allpass filter controls the degree
to which the half-sample offset property is satisfied. A Matlab
program for the procedure is given, and examples are also given
to illustrate the results. Programs are available on the Web at
http://taco.poly.edu/selesi.
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Ivan W. Selesnick (M’92) received the B.S., M.E.E.,
and Ph.D. degrees in electrical engineering in 1990,
1991, and 1996, respectively, from Rice University,
Houston, TX.
As a Ph.D. student, he received a DARPA-NDSEG
fellowship in 1991. In 1997, he was a Visiting Professor with the University of Erlangen-Nurnberg,
Nurnberg, Germany. He is currently an Assistant
Professor in the Department of Electrical and
Computer Engineering at Polytechnic University,
Brooklyn, NY. His current research interests are in
the area of digital signal processing and wavelet-based signal processing.
Dr. Selesnick’s Ph.D. dissertation received the Budd Award for Best Engineering Thesis at Rice University in 1996 and an award from the Rice-TMC
chapter of Sigma Xi. In 1997, he received an Alexander von Humboldt Award.
He received a National Science Foundation Career award in 1999. He is currently a member of the IEEE Signal Processing Theory and Methods Technical
Committee.